Calculus Examples

$lim_{x \to \infty} \frac{2 x^{- 1} + 3 x^{- 2}}{x^{- 2} + 4}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\infty$ .

$\frac{lim_{x \to \infty} 2 x^{- 1} + 3 x^{- 2}}{lim_{x \to \infty} x^{- 2} + 4}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\infty$ .

$\frac{lim_{x \to \infty} 2 x^{- 1} + lim_{x \to \infty} 3 x^{- 2}}{lim_{x \to \infty} x^{- 2} + 4}$

Step 3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to \infty} x^{- 1} + lim_{x \to \infty} 3 x^{- 2}}{lim_{x \to \infty} x^{- 2} + 4}$

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2 lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} 3 x^{- 2}}{lim_{x \to \infty} x^{- 2} + 4}$

Step 5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{2 \cdot 0 + lim_{x \to \infty} 3 x^{- 2}}{lim_{x \to \infty} x^{- 2} + 4}$

Step 6

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 \cdot 0 + 3 lim_{x \to \infty} x^{- 2}}{lim_{x \to \infty} x^{- 2} + 4}$

Step 7

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2 \cdot 0 + 3 lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} x^{- 2} + 4}$

Step 8

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{2 \cdot 0 + 3 \cdot 0}{lim_{x \to \infty} x^{- 2} + 4}$

Step 9

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\infty$ .

$\frac{2 \cdot 0 + 3 \cdot 0}{lim_{x \to \infty} x^{- 2} + lim_{x \to \infty} 4}$

Step 10

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2 \cdot 0 + 3 \cdot 0}{lim_{x \to \infty} \frac{1}{x^{2}} + lim_{x \to \infty} 4}$

Step 11

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{2 \cdot 0 + 3 \cdot 0}{0 + lim_{x \to \infty} 4}$

Step 12

Evaluate the limit.

Tap for more steps...

Step 12.1

Evaluate the limit of $4$ which is constant as $x$ approaches $\infty$ .

$\frac{2 \cdot 0 + 3 \cdot 0}{0 + 4}$

Step 12.2

Simplify the answer.

Tap for more steps...

Step 12.2.1

Cancel the common factor of $2 \cdot 0 + 3 \cdot 0$ and $0 + 4$ .

Tap for more steps...

Step 12.2.1.1

Reorder terms.

$\frac{0 \cdot 2 + 0 \cdot 3}{0 + 4}$

Step 12.2.1.2

Factor $4$ out of $0 \cdot 2$ .

$\frac{4 (0 \cdot 2) + 0 \cdot 3}{0 + 4}$

Step 12.2.1.3

Factor $4$ out of $0 \cdot 3$ .

$\frac{4 (0 \cdot 2) + 4 (0 \cdot 3)}{0 + 4}$

Step 12.2.1.4

Factor $4$ out of $4 (0 \cdot 2) + 4 (0 \cdot 3)$ .

$\frac{4 (0 \cdot 2 + 0 \cdot 3)}{0 + 4}$

Step 12.2.1.5

Cancel the common factors.

Tap for more steps...

Step 12.2.1.5.1

Factor $4$ out of $0$ .

$\frac{4 (0 \cdot 2 + 0 \cdot 3)}{4 (0) + 4}$

Step 12.2.1.5.2

Factor $4$ out of $4$ .

$\frac{4 (0 \cdot 2 + 0 \cdot 3)}{4 \cdot 0 + 4 \cdot 1}$

Step 12.2.1.5.3

Factor $4$ out of $4 \cdot 0 + 4 \cdot 1$ .

$\frac{4 (0 \cdot 2 + 0 \cdot 3)}{4 \cdot (0 + 1)}$

Step 12.2.1.5.4

Cancel the common factor.

$\frac{4 (0 \cdot 2 + 0 \cdot 3)}{4 \cdot (0 + 1)}$

Step 12.2.1.5.5

Rewrite the expression.

$\frac{0 \cdot 2 + 0 \cdot 3}{0 + 1}$

Step 12.2.2

Simplify the numerator.

Tap for more steps...

Step 12.2.2.1

Multiply $0$ by $2$ .

$\frac{0 + 0 \cdot 3}{0 + 1}$

Step 12.2.2.2

Multiply $0$ by $3$ .

$\frac{0 + 0}{0 + 1}$

Step 12.2.2.3

Add $0$ and $0$ .

$\frac{0}{0 + 1}$

Step 12.2.3

Add $0$ and $1$ .

$\frac{0}{1}$

Step 12.2.4

Divide $0$ by $1$ .

$0$